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246                      ANALYTICAL MECHANICS
198. Coefficient of Restitution.  It is found by experiment that the ratio of the impulse of restitution to the impulse of compression depends only upon the nature of the bodies in collision. The ratio, therefore, is a constant of the substances in collision. This constant is called the coefficient of restitution, and is generally denoted by the letter e. Thus
6=-|-                                      (IX)
Vi  v
Eliminating v we obtain
But (vi  tfc) and (  Vi + v%} are the velocities of the first body relative to the second, just before and just after the collision. Denoting them by V and V, respectively, we obtain
vf e=                               .             (X)
velocity after impact
relative velocity before impact
199. Resiliency.  When two bodies rebound after collision they are said to have resiliency, and the contact is called elastic contact. The coefficient of restitution is a measure of the resiliency of the colliding bodies. When e= 1 the resiliency of the colliding bodies is perfect and the contact is said to be perfectly elastic.
The coefficient of restitution cannot have a value greater than unity, as will be seen from a consideration of the transformation of energy which takes place during collision. At the beginning of the collision the bodies have a certain amount of kinetic energy which depends upon their relative